applied fracture mechanics

Fracture Mechanics Based Models of Structural and Contact FatigueFracture Mechanics Based Models of Structural and Contact Fatigue 7151the more fatigue cracks with larger radii l exist at the point the lower is the local survivalprobability p(N, x, y, z). It is reasonable to assume that the material local survival probabilityp(N, x, y, z) is a certain monotonic measure of the portion of cracks with radius l below thecritical radius l c . Therefore, p(N, x, y, z) can be represented by the expressionsp(N, x, y, z) = 1 ρ∫l c0f (N, x, y, z, l)dl i f f (0, x, y, z, l 0 ) = 0,p(N, x, y, z) =1 otherwise,(10)ρ = ρ(N, x, y, z) = ∞ ∫0f (N, x, y, z, l)dl = ρ(0, x, y, z).Obviously, the local survival probability p(N, x, y, z) is a monotonically decreasing functionof the number of loading cycles N because fatigue crack radii l tend to grow with the numberof loading cycles N.To calculate p(N, x, y, z) from (10) one can use the specific expression for f determined by (9).However, it is more convenient to modify it as followsp(N, x, y, z) = 1 ρ∫l 0c0f (0, x, y, z, l 0 )dl 0 if f(0, x, y, z, l 0 ) = 0,(11)p(N, x, y, z) =1 otherwise,where l 0c is determined by (5) and ρ is the initial volume density of cracks. Thus, toevery material point (x, y, z) is assigned a certain local survival probability p(N, x, y, z),0 ≤ p(N, x, y, z) ≤ 1.Equations (11) demonstrate that the material local survival probability p(N, x, y, z) ismainly controlled by the initial crack distribution f (0, x, y, z, l 0 ), material fatigue resistanceparameters g 0 and n, and external contact and residual stresses. Moreover, the materiallocal survival probability p(N, x, y, z) is a decreasing function of N because l 0c from (5) is adecreasing function of N for n > 2.2.6. Global fatigue damage accumulationThe survival probability P(N) of the material as a whole is determined by the localprobabilities of all points of the material at which fatigue cracks are present. It is assumedthat the material fails as soon as it fails at just one point. It is assumed that the initial crackdistribution in the material is discrete. Let p i (N) =p(N, x i , y i , z i ), i = 1,...,N c , where N cis the total number of points in the material stressed volume V at which fatigue cracks arepresent. Then based on the above assumption the material survival probability P(N) is equaltoP(N) = N c∏ p i (N). (12)i=1